Minimizing the Number of Triangular Edges

Gruslys, Vytautas; Letzter, Shoham

doi:10.1017/s0963548317000189

Cited by 4 publications

(3 citation statements)

References 8 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…Gruslys and Letzter [6] using a refined version of the symmetrization method proved that there exists an n 0 such that Tr(n, e) = g(n, e) for all n > n 0 . The second part of our Conjecture 1, namely that the extremal graph should be from a G(a, b, c), is still open.…”

Section: Further Problems Minimizing C 2k+1 Edgesmentioning

confidence: 99%

The Minimum Number of Triangular Edges and a Symmetrization Method for Multiple Graphs

Füredi

Maleki

2017

Combinator. Probab. Comp.

View full text Add to dashboard Cite

show abstract

Section: Further Problems Minimizing C 2k+1 Edgesmentioning

confidence: 99%

The Minimum Number of Triangular Edges and a Symmetrization Method for Multiple Graphs

Füredi

Maleki

2017

Combinator. Probab. Comp.

View full text Add to dashboard Cite

show abstract

“…An optimal construction comes from partitioning the vertex set into A ∪ B ∪ C and adding all edges within A as well as edges between A ∪ C and B. In this construction, the non-triangular edges are those between B and C. One should optimize over possible sizes of the partition A ∪ B ∪ C to maximize the number of non-triangular edges while maintaining the required total number of edges (that this number is optimal is proved asymptotically in [13] and exactly in [15] for sufficiently large graphs). The precise statement that we will use is given below, where the the function f (ρ) below is best possible due to the construction just described.…”

Section: 4mentioning

confidence: 99%

Enumerating k-SAT functions

Dong¹,

Mani²,

Zhao³

2021

Preprint

View full text Add to dashboard Cite

How many k-SAT functions on n boolean variables are there? What does a typical such function look like? Bollobás, Brightwell, and Leader conjectured that, for each fixed k ≥ 2, the number of k-SAT functions on n variables is (1 + o(1))2 ( n k )+n , or equivalently: a 1 − o(1) fraction of all k-SAT functions are unate, i.e., monotone after negating some variables. They proved a weaker version of the conjecture for k = 2. The conjecture was confirmed for k = 2 by Allen and k = 3 by Ilinca and Kahn.We show that the problem of enumerating k-SAT functions is equivalent to a Turán density problem for partially directed hypergraphs. Our proof uses the hypergraph container method. Furthermore, we confirm the Bollobás-Brightwell-Leader conjecture for k = 4 by solving the corresponding Turán density problem.1.2. Strategy. Our setup is closest to the work of Ilinca and Kahn [18] where they enumerated 3-SAT functions.We call such w (not necessarily unique) a witness to C i . 1Example 1.3. The 2-SAT formula {wx, wy, xz, yz} is not minimal since it is impossible to satisfy only wx and no other clause. Indeed, if we attempt to only satisfy wx, we must assign w = 1 and x = 1; to avoid satisfying wy and xz, we must assign y = 0 and z = 1, which would then lead to the final clause yz being satisfied.Every k-SAT function can be expressed as a minimal formula, but possibly more than one. We upper bound the number of minimal k-SAT formulae, which in turn upper bounds the number of k-SAT functions.To bound the number of minimal k-SAT formulae, we identify a fixed set B of non-minimal formulae, and then find an upper bound to the number of (not necessarily minimal) formulae not containing anything from B as a subformulae. We will show that, by setting B to be all non-minimal formulae on a constant number of variables, the number of B-free formulae is asymptotically the same as the number of minimal formulae.Definition 1.4 (Subformula). Given a formula G, a subformula of G is a subset of clauses of G. We say that two formulae are isomorphic if one can be obtained from the other by relabeling variables. Given another formula F , a copy of F in G is a subformula of G that is isomorphic to F . Given a set B of k-SAT formulae, we say that G is B-free if G has no copy of B for every B ∈ B.The problem of counting B-free formulae is analogous to counting F -free graphs on n vertices. A classic result by Erdős, Kleitman, and Rothschild [11] shows that almost all

show abstract

“…A less studied but still quite natural question is to maximise the number of edges that do not belong to any forbidden subgraph. Such problems in the Turán context (where we are given the order n and the size m > ex(n, H) of a graph G) were studied in [13,19,22,23]. In the Ramsey context, a problem of this type seems to have been first posed by Erdős, Rousseau, and Schelp (see [12,Page 84]).…”

Section: Introductionmentioning

confidence: 99%